The Millennium Bridge

Module 2: Synchronization and Collective Behaviour

The London Millennium Footbridge opened on 10 June 2000. On the opening day it moved sideways, unexpectedly, when crowds crossed it (dallard2001london?). The engineers found that the lateral force from pedestrians depended on the motion of the deck. They concluded that the same effect could occur on any bridge with a lateral frequency below about 1.3 Hz carrying enough pedestrians. The bridge was retrofitted with viscous and tuned mass dampers (dallard2001london?).

Strogatz et al. (2005) explained the onset with a Kuramoto-type model. Bridge wobbling and crowd synchrony are two sides of one instability, and both appear when the crowd exceeds a critical size.

Modelling assumptions

The bridge is a single lateral vibration mode, a damped harmonic oscillator.
Each pedestrian is a phase oscillator whose sideways force is periodic in the stepping phase.
Pedestrians do not interact with each other. They interact only through the bridge.

Bridge

Let $X(t)$ be the lateral displacement of the relevant mode. With $N$ pedestrians,

\[ M\ddot X + B\dot X + KX = G\sum_{i=1}^N \sin\Theta_i, \qquad \Omega_0 = \sqrt{K/M}. \tag{1}\]

$M$, $B$ and $K$ are the modal mass, damping and stiffness, and $G$ is the maximum lateral force of one pedestrian (Strogatz et al. 2005). Here $K$ is a stiffness, not the Kuramoto coupling.

Pedestrians

Pedestrian $i$ has natural stepping frequency $\Omega_i$, drawn from a density $P(\Omega)$. The bridge perturbs the gait according to

\[ \dot\Theta_i = \Omega_i + C A \sin(\Psi - \Theta_i + \alpha), \qquad i = 1, \dots, N, \tag{2}\]

where $C$ measures the sensitivity to vibrations and $\alpha$ is a phase lag. The amplitude $A$ and phase $\Psi$ of the bridge are defined by

\[ X = A\sin\Psi, \qquad \dot X = A\,\Omega_0\cos\Psi. \tag{3}\]

Strogatz et al. (2005) call Equation 2 hypothetical but testable. It has the same form as the firefly equation in Phase Locking, with a stimulus whose strength $CA$ grows with the wobble.

Feedback loop

Compare Equation 2 with the mean-field Kuramoto equation in The Order Parameter. The bridge plays the role of the mean field. Crowd coherence is measured with the usual order parameter

\[ R e^{i\Phi} = \frac{1}{N}\sum_{j=1}^N e^{i\Theta_j}. \tag{4}\]

A small wobble nudges some pedestrians into step. Their forces add coherently and drive the bridge harder, which entrains more pedestrians. For small crowds, $R$ fluctuates at the level $N^{-1/2}$ and the bridge barely moves (Strogatz et al. 2005).

Critical crowd size

For $\alpha = \pi/2$ and $P(\Omega)$ symmetric about $\Omega_0$, Strogatz et al. (2005) obtain

\[ N_c = \frac{4\zeta}{\pi}\,\frac{K}{G\,C\,P(\Omega_0)}, \qquad \zeta = \frac{B}{\sqrt{4MK}}, \tag{5}\]

where $\zeta$ is the damping ratio. Since $4\zeta K = 2B\sqrt{K/M} = 2B\Omega_0$, the same result reads

\[ N_c = \frac{2B\,\Omega_0}{\pi\,G\,C\,P(\Omega_0)}. \tag{6}\]

Compare with $K_c = 2/(\pi g(0))$ in Critical Coupling. More damping raises the threshold. A narrower spread of gait frequencies (larger $P(\Omega_0)$) lowers it. Strogatz et al. (2005) estimated $C \approx 16\ \text{m}^{-1}\text{s}^{-1}$ from crowd tests on the bridge. Eckhardt et al. (2007) analyse the model in more detail.

Exercise

Check that Equation 6 is dimensionless. Use $[G] = \text{N}$, $[B] = \text{kg/s}$, $[C] = \text{m}^{-1}\text{s}^{-1}$ and $[P] = \text{s}$ (a density over angular frequency).

Solution

The numerator $B\Omega_0$ has units $\text{kg}\,\text{s}^{-2}$. The denominator $GCP$ has units $(\text{kg}\,\text{m}\,\text{s}^{-2})(\text{m}^{-1}\text{s}^{-1})(\text{s}) = \text{kg}\,\text{s}^{-2}$. The ratio is dimensionless, as a number of pedestrians should be.

Exercise

Take $P(\Omega)$ normal with mean $\Omega_0$ and standard deviation $\sigma$. Write $N_c$ in terms of $\sigma$. How does $N_c$ change if the damping is doubled?

Solution

$P(\Omega_0) = 1/(\sqrt{2\pi}\,\sigma)$, so $N_c = 2\sqrt{2\pi}\,B\,\Omega_0\,\sigma/(\pi G C)$. $N_c$ is proportional to $B$, so doubling the damping doubles the critical crowd size. This is the reasoning behind adding dampers.

You will simulate this model in the graded Kuramoto Assignment.

References

Eckhardt, Bruno, Edward Ott, Steven H. Strogatz, Daniel M. Abrams, and Alan McRobie. 2007. “Modeling Walker Synchronization on the Millennium Bridge.” Physical Review E 75: 021110.

Strogatz, Steven H., Daniel M. Abrams, Alan McRobie, Bruno Eckhardt, and Edward Ott. 2005. “Crowd Synchrony on the Millennium Bridge.” Nature 438: 43–44.

--- title: "The Millennium Bridge" subtitle: "Module 2: Synchronization and Collective Behaviour" format: html --- The London Millennium Footbridge opened on 10 June 2000. On the opening day it moved sideways, unexpectedly, when crowds crossed it [@dallard2001london]. The engineers found that the lateral force from pedestrians depended on the motion of the deck. They concluded that the same effect could occur on any bridge with a lateral frequency below about 1.3 Hz carrying enough pedestrians. The bridge was retrofitted with viscous and tuned mass dampers [@dallard2001london]. @strogatz2005bridge explained the onset with a Kuramoto-type model. Bridge wobbling and crowd synchrony are two sides of one instability, and both appear when the crowd exceeds a critical size. ## Modelling assumptions - The bridge is a single lateral vibration mode, a damped harmonic oscillator. - Each pedestrian is a phase oscillator whose sideways force is periodic in the stepping phase. - Pedestrians do not interact with each other. They interact only through the bridge. ## Bridge Let $X(t)$ be the lateral displacement of the relevant mode. With $N$ pedestrians, $$ M\ddot X + B\dot X + KX = G\sum_{i=1}^N \sin\Theta_i, \qquad \Omega_0 = \sqrt{K/M}. $$ {#eq-bridge} $M$, $B$ and $K$ are the modal mass, damping and stiffness, and $G$ is the maximum lateral force of one pedestrian [@strogatz2005bridge]. Here $K$ is a stiffness, not the Kuramoto coupling. ## Pedestrians Pedestrian $i$ has natural stepping frequency $\Omega_i$, drawn from a density $P(\Omega)$. The bridge perturbs the gait according to $$ \dot\Theta_i = \Omega_i + C A \sin(\Psi - \Theta_i + \alpha), \qquad i = 1, \dots, N, $$ {#eq-pedestrian} where $C$ measures the sensitivity to vibrations and $\alpha$ is a phase lag. The amplitude $A$ and phase $\Psi$ of the bridge are defined by $$ X = A\sin\Psi, \qquad \dot X = A\,\Omega_0\cos\Psi. $$ {#eq-amp-phase} @strogatz2005bridge call @eq-pedestrian hypothetical but testable. It has the same form as the firefly equation in [Phase Locking](phase-locking.qmd), with a stimulus whose strength $CA$ grows with the wobble. ## Feedback loop Compare @eq-pedestrian with the mean-field Kuramoto equation in [The Order Parameter](order-parameter.qmd). The bridge plays the role of the mean field. Crowd coherence is measured with the usual order parameter $$ R e^{i\Phi} = \frac{1}{N}\sum_{j=1}^N e^{i\Theta_j}. $$ {#eq-bridge-order} A small wobble nudges some pedestrians into step. Their forces add coherently and drive the bridge harder, which entrains more pedestrians. For small crowds, $R$ fluctuates at the level $N^{-1/2}$ and the bridge barely moves [@strogatz2005bridge]. ## Critical crowd size For $\alpha = \pi/2$ and $P(\Omega)$ symmetric about $\Omega_0$, @strogatz2005bridge obtain $$ N_c = \frac{4\zeta}{\pi}\,\frac{K}{G\,C\,P(\Omega_0)}, \qquad \zeta = \frac{B}{\sqrt{4MK}}, $$ {#eq-nc} where $\zeta$ is the damping ratio. Since $4\zeta K = 2B\sqrt{K/M} = 2B\Omega_0$, the same result reads $$ N_c = \frac{2B\,\Omega_0}{\pi\,G\,C\,P(\Omega_0)}. $$ {#eq-nc-b} Compare with $K_c = 2/(\pi g(0))$ in [Critical Coupling](critical-coupling.qmd). More damping raises the threshold. A narrower spread of gait frequencies (larger $P(\Omega_0)$) lowers it. @strogatz2005bridge estimated $C \approx 16\ \text{m}^{-1}\text{s}^{-1}$ from crowd tests on the bridge. @eckhardt2007modeling analyse the model in more detail. ::: {.callout-note} ## Exercise Check that @eq-nc-b is dimensionless. Use $[G] = \text{N}$, $[B] = \text{kg/s}$, $[C] = \text{m}^{-1}\text{s}^{-1}$ and $[P] = \text{s}$ (a density over angular frequency). ::: ::: {.callout-tip collapse="true"} ## Solution The numerator $B\Omega_0$ has units $\text{kg}\,\text{s}^{-2}$. The denominator $GCP$ has units $(\text{kg}\,\text{m}\,\text{s}^{-2})(\text{m}^{-1}\text{s}^{-1})(\text{s}) = \text{kg}\,\text{s}^{-2}$. The ratio is dimensionless, as a number of pedestrians should be. ::: ::: {.callout-note} ## Exercise Take $P(\Omega)$ normal with mean $\Omega_0$ and standard deviation $\sigma$. Write $N_c$ in terms of $\sigma$. How does $N_c$ change if the damping is doubled? ::: ::: {.callout-tip collapse="true"} ## Solution $P(\Omega_0) = 1/(\sqrt{2\pi}\,\sigma)$, so $N_c = 2\sqrt{2\pi}\,B\,\Omega_0\,\sigma/(\pi G C)$. $N_c$ is proportional to $B$, so doubling the damping doubles the critical crowd size. This is the reasoning behind adding dampers. ::: You will simulate this model in the graded [Kuramoto Assignment](assignment-kuramoto.qmd).